For example, let us consider a TTUR based on stochastic gradient descent (SGD) [1]}, [2]}, [3]}; let \(\mathbf {\theta }_n\)  be the point generated by the generator at iteration \(n\) , and \(\mathbf {w}_n\)  the point generated by the discriminator at iteration \(n\) . The generator uses SGD with a learning rate \(\alpha _n^G\)  and a mini-batch stochastic gradient at \(\mathbf {\theta }_n\)  using a batch of size \(b\)  for minimizing its loss function \(L_G(\cdot , \mathbf {w}_n)\) . Meanwhile, the discriminator uses SGD with a learning rate \(\alpha _n^D\)  and a mini-batch stochastic gradient at \(\mathbf {w}_n\)  using a batch of size \(b\)  for minimizing its loss function \(L_D(\mathbf {\theta }_n, \cdot )\) . If \(\alpha _n^G\)  and \(\alpha _n^D\)  are decaying learning rates, then TTUR based on SGD converges almost surely to a stationary local Nash equilibrium [4]}.
